This page intentionally left blank Linear Programming and Network Flows This page intentionally left blank Linear Programming and Network Flows Fourth Edition Mokhtar S Bazaraa Agility Logistics Atlanta, Georgia John J Jarvis Georgia Institute of Technology School of Industrial and Systems Engineering Atlanta, Georgia HanifD.Sherali Virginia Polytechnic Institute and State University Grado Department of Industrial and Systems Engineering Blacksburg, Virginia ©WILEY A John Wiley & Sons, Inc., Publication Copyright © 2010 by John Wiley & Sons, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose No warranty may be created or extended by sales representatives or written sales materials The advice and strategies contained herein may not be suitable for your situation You should consult with a professional where appropriate Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic format For information about Wiley products, visit our web site at www.wiley.com Library of Congress Cataloging-in-Publication Data: Bazaraa, M S Linear programming and network flows / Mokhtar S Bazaraa, John J Jarvis, Hanif D Sherali — 4th ed p cm Includes bibliographical references and index ISBN 978-0-470-46272-0 (cloth) Linear programming Network analysis (Planning) I Jarvis, John J II Sherali, Hanif D., 1952- III Title T57.74.B39 2010 519.7'2—dc22 2009028769 Printed in the United States of America 10 Dedicated to Our Parents This page intentionally left blank CONTENTS Preface xi ONE: INTRODUCTION 1.1 The Linear Programming Problem 1.2 Linear Programming Modeling and Examples 1.3 Geometric Solution 1.4 The Requirement Space 1.5 Notation Exercises Notes and References TWO: LINEAR ALGEBRA, CONVEX ANALYSIS, AND POLYHEDRAL SETS 2.1 Vectors 2.2 Matrices 2.3 Simultaneous Linear Equations 2.4 Convex Sets and Convex Functions 2.5 Polyhedral Sets and Polyhedral Cones 2.6 Extreme Points, Faces, Directions, and Extreme Directions of Polyhedral Sets: Geometric Insights 2.7 Representation of Polyhedral Sets Exercises Notes and References THREE: FOUR: FIVE: 1 18 22 27 29 42 45 45 51 61 64 70 71 75 82 90 THE SIMPLEX METHOD 3.1 Extreme Points and Optimality 3.2 Basic Feasible Solutions 3.3 Key to the Simplex Method 3.4 Geometric Motivation of the Simplex Method 3.5 Algebra of the Simplex Method 3.6 Termination: Optimality and Unboundedness 3.7 The Simplex Method 3.8 The Simplex Method in Tableau Format 3.9 Block Pivoting Exercises Notes and References 91 91 94 103 104 108 114 120 125 134 135 148 STARTING SOLUTION AND CONVERGENCE 151 SPECIAL SIMPLEX IMPLEMENTATIONS AND OPTIMALITY CONDITIONS 5.1 The Revised Simplex Method 5.2 The Simplex Method for Bounded Variables 201 201 220 4.1 4.2 4.3 4.4 4.5 4.6 4.7 The Initial Basic Feasible Solution The Two-Phase Method The Big-MMethod How Big Should Big-WBe? The Single Artificial Variable Technique Degeneracy, Cycling, and Stalling Validation of Cycling Prevention Rules Exercises Notes and References 151 154 165 172 173 175 182 187 198 vii viii Contents 5.3 5.4 SIX: SEVEN: EIGHT: NINE: Farkas' Lemma via the Simplex Method The Karush-Kuhn-Tucker Optimality Conditions Exercises Notes and References DUALITY AND SENSITIVITY ANALYSIS 6.1 Formulation of the Dual Problem 6.2 Primal-Dual Relationships 6.3 Economic Interpretation of the Dual 6.4 The Dual Simplex Method 6.5 The Primal-Dual Method 6.6 Finding an Initial Dual Feasible Solution: The Artificial Constraint Technique 6.7 Sensitivity Analysis 6.8 Parametric Analysis Exercises Notes and References THE DECOMPOSITION PRINCIPLE 7.1 The Decomposition Algorithm 7.2 Numerical Example 7.3 Getting Started 7.4 The Case of an Unbounded Region X 7.5 Block Diagonal or Angular Structure 7.6 Duality and Relationships with other Decomposition Procedures Exercises Notes and References COMPLEXITY OF THE SIMPLEX ALGORITHM AND POLYNOMIAL-TIME ALGORITHMS 8.1 Polynomial Complexity Issues 8.2 Computational Complexity of the Simplex Algorithm 8.3 Khachian's Ellipsoid Algorithm 8.4 Karmarkar's Projective Algorithm 8.5 Analysis of Karmarkar's Algorithm: Convergence, Complexity, Sliding Objective Method, and Basic Optimal Solutions 8.6 Affine Scaling, Primal-Dual Path Following, and Predictor-Corrector Variants of Interior Point Methods Exercises Notes and References MINIMAL-COST NETWORK FLOWS 9.1 The Minimal Cost Network Flow Problem 9.2 Some Basic Definitions and Terminology from Graph Theory 9.3 Properties of the A Matrix 9.4 Representation of a Nonbasic Vector in Terms of the Basic Vectors 9.5 The Simplex Method for Network Flow Problems 9.6 An Example of the Network Simplex Method 9.7 Finding an Initial Basic Feasible Solution 9.8 Network Flows with Lower and Upper Bounds 234 237 243 256 259 259 264 270 277 285 293 295 312 319 336 339 340 345 353 354 361 371 376 391 393 393 397 401 402 417 428 435 448 453 453 455 459 465 466 475 475 478 734 assignment problem, 535 capital budgeting, 11 copper market, 510 coal blending and distribution, 15, 36 cutting stock, 10, 191,384 discrete control problem, 386 distribution problem, 37 equilibrium problems, 38, 324 facility location problem, 30 feed-mix, 8, 29 game theory problems, 41, 324 housing renewal planning problem, 34 investment problem, 29, 31, 34 maximal flow problem, 607 menu planning problem, 30 minimal-cost maximal flow problem, 672 minimal-cost network flow problem, 453 multicommodity flow problem, 639 personnel training problem, 30 production scheduling, 9, 31, 33 production inventory, 37 production-transportationinventory problem, 505 project management problem, 669 project selection problem, 34 rocket launching problem, 35 scheduling problems, 9, 31, 32, 33, 36 tanker scheduling, 13 tourism problem, 332 traffic assignment problem, 37, 675 transportation problem, 11, 32, 513 two-stage stochastic problem with recourse, 39 warehouse location problem, 388 Approach-dependent shortest path problem, 639, 679 Arborescence, 456, 622, 631 Arc: capacities, 608 definition, 453 in-kilter/out-of-kilter, 561 label eligible, 569 one-arcs, 545 zero-arcs, 539 Arc disjoint paths, 664 Arc-path formulation, 608, 666 Artificial intelligence modeling approach, 42 Artificial technique: Index single constraint, 293 single variable, 173 Artificial variable, 153, 198, 522 Assignment problem: alternating basis, 544, 564 covering, 540, 560, 666 definition, 535 dual problem, 537 finite convergence, 543, 550 independent cells, 540, 560, 666 (Kuhn's) Hungarian algorithm modifying dual variables, 540 partial solution, 540 polytope, 537, 561, 564 problem, 535 reduced matrix, 538 successive shortest path, 546 Assumptions in linear programming, Augmented matrix, 56, 61 Augmented threaded index, See also Data structures, 482 Average-case complexity, 449 Back substitution, 56 Balanced transportation problem, 514 Barrier function, 444 Barrier problem, 431 Basic feasible partition, 222 Basic feasible solution: bounded variables, 222 definition, 62, 94, 95 degenerate, 95, 98, 222 existence, 101 improving, 111, 223 initial, 129, 151, 293, 475, 489, 522 nondegenerate, 95, 222 number of, 97, 224 optimal, 21, 104,225 relationship to extreme points, 99 Basic dual solution, 282 Basic solution, 62, 95 Basic variables, 95, 222 Basis: adjacent, 106 alternating path, 544, 564 assignment problem, 561 compact, 221 complementary dual, 282 crash, 173 definition, 49, 61, 95 entry criterion, 106, 111, 121, 178, 180,225,493 exit criterion, 112, 121, 178, 180, 479, 527 Index generalized network, 494, 512 Karmarkar's algorithm, 424 matrix, 61, 95 maximal flow problem, 613 multicommodity flow problem, 649 network flow problem, 461, 504 number of, 97, 224 optimal, 21, 104,225 relationship to trees in networks, 461,463,518 transportation, 518 triangularity of in networks, 463, 518 working, 221 Basis equivalent chain, 465, 469 Benders' partitioning procedure, 371, 372, 388 Bidirectional search, 452 Big-M method: analysis, 167 comparison with two-phase method, 168 description, 165, 172 Binary encoding, 394 Binding constraints, active, tight, 71 Bipartite graph, 514, 536 Bland's cycling prevention rule, 180, 184,198,230 Block diagonal constraint matrix, 361, 362 Blocking hyperplane, constraint, 106 Blocking variable, 106, 112 Block pivoting, 134, 599, 613 Bounded set, 20, 75 Bounded variable, 220, 478 Bounded variables affine scaling algorithms, 446 Boxstep method, 374, 390, 392 Branch-and-price, 392 Breakthrough, 576, 586, 594 BTRAN, 209 Bumps, 219 Busacker-Gowen algorithm, 672 Canonical form: of duality, 259 of linear program, 5, 71, 104 of simplex tableau, 126 Capacitated network, 454 Capacitated transportation problem, 559 Capacity: arc, 478, 608 cut, 609 735 disconnecting set, 676 forward cut, 609 Capital budgeting problem, 11 Caratheodory theorem, 77 Central path, 432 Chain: basis equivalent, 465, 469 in graph, 456 in transportation tableau, 517 Change: in basis, 50, 109 of constraint coefficients, 298 of cost vector, 296 of right-hand-side, 297 Chord, 70 Circuit, 456, 634 Circulatory network flow problem, 568 Circulation, 568 Class P of problems, 396 Closed chain, 456 Closed interval, 29 Closed path, 456 Closed system, 439 Cofactor, 59 Column generation scheme, 339, 343, 372, 657, 658, 680 Column generation, stabilized, 344, 374,391,392 Column pivoting, 250 Column rank, 61 Column simplex method, 250 Column vector, 45 Combination: convex, 64 linear, 48 Communication network, 654 Compact basis, 221 Compact form, 214 Comparison: of simplex and revised simplex, 205 Complementarity theorem, 327 Complementary basic dual solution, 282 pair of variables, 269 slackness conditions, 239, 268, 538, 569 slackness theorems, 268, 336 Complete graph, 456, 514 Complexity: average-case, 449 computational, definition, 81, 394 genuinely (strongly) polynomial, 396,450,617 736 of Karmarkar's algorithm, 418 of Khachian's algorithm, 437 order, 394 polynomial, 394 ofPSP algorithm, 636 of shortest path problem, 617, 622, 633 of simplex method, 397 strongly (genuinely) polynomial, 396,450,617 Complicating constraints, 339 Complicating variables, 371 Computational complexity, 394, 397, 418 Component of graph, 456 Concave function, 70 Cone: convex, 68, 237 generated by vectors, 69 polyhedral, 71 recession, 74 Connected graph: strongly, 456 weakly, 456 Conserving flow, 567, 595 Constraint: active, binding, tight, 71 artificial, 293 complicating, 339 coupling, 386 definition, functional, matrix, nonnegativity, Construction of basic feasible solutions, 94 Consumption column, 301 Control variable, Convergence: assignment algorithms, 543, 552 bounded variable simplex method, 229 dual simplex method, 285 interior point methods, 429 Karmarkar's algorithm, 418 Khachian's algorithm, 439 maximal flow algorithm, 611 out-of-kilter algorithm, 585, 591 primal-dual method, 292 shortest path algorithms, 622, 630, 633 simplex method, 122, 181 Convex: analysis, 45 Index arc costs, 605 combination, 64 cone, 68 function, 64, 70 set, 64 Convexity constraint, 362 Coordinate ascent, 605 Coordinate vector, 46 Corner point, 72 Corners of cycle in transportation tableau, 517, 520 Corrector step, 435 Cost coefficient, Court scheduling problem, 36 Covering in assignment problem, 540, 560, 666 CPLEX, 511,605 Cramer's Rule, 60 Criss-cross algorithm, 333 Criterion function, Critical path problem, 600, 669 Cut, 305, 372, 609 Cut-set, 609 Cutting plane algorithms, 305, 372 Cutting stock problem, 10, 191, 384 Cycle: in graph, 456, 575, 634 in transportation tableau, 517, 520 Cycle method for computing z ,·, - c,·,·, 525, 564 Cycling: example, 175 geometry, 177, 199, 337 in networks, 482, 509 phenomenon, 106,229 practical prevention rule, 180, 198 prevention rules, 178 validation of prevention rules, 182 Dantzig's Rule, 121 Dantzig-Wolfe decomposition method, 340, 642, 680 Data structures, for network flows: antecedents, 483 augmented threaded index, 482 down, 482 DUAL vector, 485 final of node, 484 grafting, 486 FLOW vector, 485 immediate successor, 483 last successor, 484 list structures, 482 level, 482 Index lower tree, 488 next, 483 ORIENT vector, 484 postorder traversal, 483 predecessor, 482 preorder distances, 483 preorder traversal, 483 reverse thread, 483 subtree rooted at node, 482 successors, 482 thread, 482 upper tree, 488 Decision problem, 396, 436 Decision variables, Decomposition algorithm: algorithm/principle, 339 block diagonal structure, 361 economic interpretation, 369 getting started, 353 lower bound on objective function, 345, 364 multicommodity network flow problem, 641 nested, 384, 385 network synthesis problem, 607, 654, 680 unbounded subproblem region, 354 Defining: hyperplane, 72 variable, 104 Degeneracy: in assignment problem, 537 in basic feasible solutions, 72, 95, 98,221,222,229 in networks, 488 order, 72 relationship to cycling, 175 to shadow prices, 273 in transportation problems, 523, 528,531 Degenerate iteration/pivot, 106, 229, 230 Degree of node: definition, 457 in-degree, 457 out-degree, 457 Degrees of freedom, 104 Density, 206 Dependent: constraints, 61 variables, 95, 221 vectors, 49 737 Destination in transportation problems, 513 Determinant of matrix, 59 Deterministic assumption, Digraph, 453, 455 Dijkstra's algorithm, 620 Dimension: of basis, 50 of Euclidean space, 48 full, 73 of matrix, 51 of set (region), 73 of vector, 45 Directed: arc, 453 cycle, 456 network, 453 Direction: associated with unbounded solution, 118 of convex set, 66 distinct, 68 extreme, 68, 71 of polyhedral set, 66 of ray, 66, 118 recession, 66, 71 Disconnecting an arc, 457 Disconnecting set, 676 Discrete control problem, 386 Discrete optimization, Distribution problem, 38 Divisibility assumption, Dominant requirement tree, 655, 656 Dot product, 47 Dual: affine scaling, 431 angular structure, 384 canonical form, 259 complementary basis, 282 feasibility and primal optimality, 239, 243 formulation, 259 mixed forms, 262 of assignment problem, 537 of circulatory network-flow problems, 569 of dual, 261 of maximal flow problem, 610 of out-of-kilter formulation, 569 phase, 577, 587 problem, 239 relationship with primal problem, 264 simplex method, 279 738 standard form, 260 variables, 239, 260, 269 Duality: Fundamental theorem, 267 and Karush-Kuhn-Tucker conditions, 265 and Lagrangian multipliers, 239, 243,371 economic interpretation, 270 gap, 324, 592 involutory property, 262 origin, 265 strong, 266 supervisor's principle, 268 theorems, 267, 268 weak, 264 Dual feasibility, 239, 278 Dual simplex method: bounded variables, 333 development, 277, 499 finite convergence, 285 getting started, 293 summary, 279, 281 Dual variable method, 525 Dynamic shortest path problems, 679 Dynamic stochastic shortest path problems, 639, 679 Dynamic trees, 512 Economic interpretation: of decomposition, 369 of duality, 270 Edge of polyhedron, 73 of graph, 454 Efficient solutions, Elementary matrix, 207 Elementary matrix operations, 54 Elimination form of the inverse, 256 Empty feasible region, 22 Encoding: binary, 394 stroke, 396, 617 End node of a tree, 457, 518 Entry criterion, 106, 111, 121, 178, 180,225,493 Epigraph, 87 Equal flow constraint, 511 Equality constrained polyhedron, 82 Equilibrium, 39 Equivalent weights, 199 Eta vector, 208 Euclidean norm, 47 Euclidean space, 48 Index Excess flow, 679 Excess function, 594 Exit criterion, 112, 121, 178, 180,479, 527 Extreme direction, 68, 71, 74 Extreme point: adjacent, 73, 254 definition, 64, 71,72 optimality at, 91, 114 relationship to basic feasible solutions, 99 representation theorem, 75, 76 Extract a node/arc, 628, 633, 635 Extreme ray, 68, 71, 74 Face: improper, 73 proper, 71, 73 Facility location, 30 Factorization: interior point methods, 450 LU, 212 QR, 443 Fair market price, 271, 273 Farkas' Lemma, 234, 235 Feasible: flow, 567 region/space, solution, system, 23 Feed mix problem, 8, 29 Fibonacci heaps, 679 Final node, 484 Finite convergence, see Convergence Finite optimal solution, 91, 114 First-in-first-out, 635, 638 Flow: in arc, 454 augmentation, 548, 568 conservation equations, 454 excess, 679 maximal, 607 minimal-cost, 454 multicommodity, 639 with gains, 510, 512 Forest graph, 457 Forward arcs of a cut, 609 Forward cut, 609 Forward-star, 455, 628 Fourier-Motzkin elimination method, 144 Fractional part, 304 From-node, 455 FTRAN, 209 Index Full dimensional, 73 Full rank matrix, 61 Functional constraint, Fundamental theorem of duality, 267 Game theory, 42, 324 Gaussian reduction, 56, 63, 214 Gaussian reduction matrix, 214 Gaussian triangularization, 212 Gauss-Jordan reduction, 56, 62, 212 Generalized linear programming problem, 385 Generalized networks, 512, 494 Generalized transportation problem, 558 Generalized upper bounding, 257 General solution of linear equations, 62 Genuinely (strongly) polynomial, 396, 450, 617 Geometric interpretation: Farkas' lemma, 236 Karush-Kuhn-Tucker conditions, 237 Geometric redundancy, 71, 190 Geometric solution of linear programs, 18, 104 Geometry of cycling, 177, 199, 337 Gomory's dual fractional cut, 305 Gradient, 29, 65 Grafting, 486 Graph: bipartite, 514, 536 complete, 456, 514 component, 456 connected, strongly, weakly, 456 definition, 455 digraph, 455 forest, 457 mixed, 455 proper, 455 tree, 456 undirected, 455 Half line, 69 Half-space, 65, 66 Heap Dijkstra procedure, 633 Heap implementation, 633, 637, 679 Hidden networks, 512 Hirsch conjecture, 449, 564 Hitchcock (transportation) problem, 42, 514 Homogeneous system, 74, 267 Housing renewal planning problem, 34 739 Hungarian (Kuhn's) method, 535, 541, 564 Hyperplane: definition, 65 normal (gradient) to, 65 Hypograph, 87 Identity matrix, 52 Immediate successor, 483 Improving basic feasible solutions, 111,223 Inputed values, 273 Inactive constraint, 71 Incident, 454 Inconsistent system, 21, 22, 190 Incremental cost, 271 In-degree of node, 457 Independent cells in assignment problem, 540, 560, 666 Independent variables, 95, 221 Independent vectors, 48 Induced by node set, 456 Infeasible, 21 Initial basic feasible solution, 129, 151, 293, 475, 489, 522 In-kilter, 570 Inner optimization problem, 371 Inner product, 47 Input length of problem, 394 Input-output management, 259 Insert operations, 633 Instance of problem, 394 Integer part of coefficient, 304 Integer programming problem, 304, 536 Integer property: in assignment problems, 536 in network flow problems, 463 in transportation problems, 517 Integer variable, 304 Interior point methods, 429, 605 Intermediate node, 454 Interval: closed, 29 open, 29 of uncertainty, 397 Inverse matrix: calculation, 56 condition for existence, 56 definition, 56 from simplex tableau, 132 product form, 140,207 Investment problem, 29, 31, 34 Involutory property, 262 Irreducible infeasible system, 43 740 Irrelevant constraint, 71 Iteration, 106, 109,473 Karmarkar's (projective) algorithm: complexity analysis, 418 convergence, 418 description, 402 determining an optimal basis, 424 form of linear program, 414 potential function, 420 sliding objective method, 424 Karush-Kuhn-Tucker conditions: for equality constraints, 241 geometric interpretation, 237 for inequality constraints, 237 optimality conditions, 265 perturbed, 432 proof, 237 relationship to: duality, 265 simplex method, 242 Key for heaps, 633 Khachian's algorithm, 401, 437, 450 Kilter: number, 571 state, 571,572 Kirchhoff equations, 454 Klee-Minty polytope, 398, 399 Klein's algorithm, 672 Knapsack problem, 138, 246 Kuhn's Hungarian algorithm, 537 Label-correcting algorithm, 628 Label eligible arc, 576 Labeling algorithm: maximal flow, 613 network simplex, 482 out-of-kilter, 589 shortest path, 633 Label-constrained shortest path problem, 679 Labeling method, 607 Label-setting algorithm, 621, 624, 628 Lagrangian dual problem, 335, 374 Lagrangian multipliers, 239, 243, 371 Lagrangian relaxation, 371, 373 Lagrangian subproblem, 374 Last successor, 484 Leading node, 474 Leaf node, 457 Least-recently-considered rule, 181, 491 Least squares problem, 407 Leaving variable, 106 Index Left-hand shadow price, 275 Legitimate variables, 153 Length of a stage, 181 Leontief input-output model, 42 Level index, 482 Lexicographically nonnegative vector, 182 Lexicographic ordering, 198 Lexicographically positive vector, 182 Lexicographic cycling prevention rule, 178,285 Linear: dependence, 49 independence, 48 subspace, 48 Linear combination, 48 Linear equations: basic solution, 62 Gaussian reduction, 56, 63 general solution, 62 number of solutions, 62 redundant, dependent, 61 Linear fractional program, 404 Linear inequalities, 2, Linear programming problem: assumptions in, canonical form, 5, examples, formulation, generalized, 385 geometry, 18 standard form, 5, Linear subspace, 48 Linear transformation, 404, 652 Line belonging to set, 90 Line search problem, 604 Line segment, 64 Link (arc), 453 List structures, see Data structures Longest path problem, 669 Low-order polynomial bound, 395 Lower bounds: on objective function, 345, 364 on variables, 220, 478, 655 Lower tree, 488 Lower triangular matrix, 53 LU decomposition/factorization, 212 Machine location problem, 29 Machine scheduling problem, 31, 33 Manhattan distance, 30 Manipulation of linear program, Marginal cost, 471 Master array, 342 Index Master Problem, 339, 340, 355, 362, 371 Matching, 536 Mathematical model, Matrix: addition, 52 adjoint, 60 augmented, 56 definition, 51 determinant, 59 diagonal, 58 elementary, 207 Gaussian reduction, 56, 63, 214 generator, 220 identity, 52 inverse, 56, 63 multiplication, 46, 52 multiplication by a scalar, 52 nonsingular, 56 operations, 52 partitioned, 53 permutation, 214 pivot, 214 positive definite, 438 postmultiplying, 208 premultiplying, 208 rank, 61 singular, 56 skew-symmetric, 53, 325 symmetric, 53 transpose, 53 triangular, 53 upper Hessenberg, 218 of vectors, 46 zero, 52 Matrix minimum method, 558 Maximal flow problem: algorithm, 611,612, 678 basic solutions, 613 connected subgraphs, 456 cuts, 609 dual problem, 610 formulation, 607 max flow-min cut theorem, 612, 678 multicommodity, 639, 676 scaling algorithm, 617 Maximal flow realizable network, 677 Maximum spanning tree, 656 Menu planning problem, 30 Minimal-cost flow problem: algorithm, 474, 478 basic characterization, 461, 504 formulation, 453 741 initial solution 475 lower-upper bounds on arc flows, 478 simplex tableau, 481 Minimal-cost-maximal flow problem, 672 Minimal capacity, 661, 676 Minimal cut, 609 Minimal forward cut, 676 Minimum ratio test, 121, 280 Minimum weighted matching problem, 536 Modeling, Modeling languages, 257 Multicommodity: basis characterization, 649 decomposition algorithm, 641 maximal flow problem, 640, 680 minimal-cost flow problem, 639, 640, 642 minimal disconnecting set, 676 transportation problem, 381 Multi-criteria shortest path problem, 679 Multiobjective program, 195 Multiterminal network, 657 Negative cost circuit, 632, 634 Nested decomposition method, 384, 385 NETGEN, 511,605 NETOPT, 511,605 Network analysis, 654 Network design, 654 Network, see Graph circulatory, 568 connected, 456 directed, 453 generalized 494, 512 hidden, 512 Network flow problem, 453, 567, 608, 620 Network flow with gains problem, 494, 512,558 Network simplex algorithm: computing: basic solutions, 466 dual variables, 469 determination: entering variable, 469 exit variable, 472 initial basic feasible solution, 475 labeling algorithm for, 482 list structures for, 482 742 lower-upper bounds, 478 pivoting, 472 tableau associated with, 481 Network synthesis problem, 607, 654, 680 New activity, 301 NEXT list, 635 Next node, 483 Nodal balance, 454 Node: adjacent, 455 capacitated, 501 definition, 453 end, 457 from-, 455 intermediate, 454 leading, 474 leaf, 457 potential, 470 rank (degree), 457 root, 461 to-, 455 transshipment, 454 Node-arc formulation: maximal flow problem, 608 multicommodity minimal-cost flow problem, 641 Node-arc incidence matrix, 459 Nonadjacent extreme point methods, 149 Nonbasic matrix, 61, 95 variables, 95 variable-space, 104 Nonbinding constraint, 71 Nonbreakthrough, 577, 586, 594 Nonconvex, 64 Nondegeneracy, 95, 222, 557 Nondegenerate basic feasible solution, 95, 222 Nondegenerate iteration/pivot, 107 Nonnegativity constraints, Nonsimple path, 456 Nonsingular matrix, 56 Normal to hyperplane, 48, 65 Norm of vector, 47 Northwest corner rule, 522 Notation, 27 NOW list, 635 NP-complete, 616 Null space, 407 Number of basic feasible solutions, 97, 224 Index Objective contour, 19 Objective function: definition, parametric, 312 Phase I, 154 Phase II, 155 piecewise-linear and concave, 315 piecewise-linear and convex, 318, 503 unbounded optimal value, 21, 27, 117,265,267,280,288 value, One-arc, 545 Open halfspace, 237 Open interval, 29 Open set, 87 Optimal (basic feasible) solution, 21, 104, 225 Optimal control problem, Optimal extreme point, 91, 114 Optimality conditions/criterion, 18, 21, 25, 114, 237, 242, 280, 288 Optimal location problem, 29 Optimal rounding procedures, 409, 410,439 Optimal solution set, 20 Optimality gap, 618 Optimization vs decision problems, 396, 436 Oracle, 295 Origin, 46 Origin-destination matrix, 675 Origin in transportation problems, 513 Orthogonal, 48 Out-degree of node, 457 Outer linearization, 374 Out-of-kilter algorithm: algorithm, 573, 586, 605 arc, 570 dual of, 569 dual variable change, 577 finite convergence, 585, 591 flow change, 574 formulation, 567 kilter number and states, 571, 572 Packed form, 206, 257 Parallel computations, 256 Parametric analysis of cost vector, 312 of right-hand-side vector, 313 shadow prices, 318 Pareto-optimal solution, 8, 679 Partial pricing, 206, 220, 256 Index Partition: basic feasible, 222 strongly feasible basis, 182, 229, 488,491,564 Partition matrix, 53 Partitioned shortest path algorithm, 635 Partitioning method, 371 Path-following algorithms, 451 Path in graph: closed, 456 definition, 455 nonsimple, 456 simple, 456 Payoff matrix, 324 Penalty function, 591 Perceptron-like algorithm, 452 Perfect competition equilibrium, 38, 324 Performance guarantee, 393 Permutation matrix, 214 Permutation structure, 178, 198 Perpendicular, 48 Persistency, Personnel training problem, 30 Perturbation: of cost vector, 312 of right-hand-side vector, 315 Perturbation method, 196, 328, 557 Perturbed KKT conditions, 432 Phase I method, 198 Phase I problem, 155 Phase II problem, 155 Piecewise-linear objective function, 315,318,503 Pivot: block, 134,599,613 column, 250 definition, 106, 127 element, 127 matrix, 214 Player: column, 324 row, 324 Pointing toward root, 488 Polyhedral: cone, 70, 71 set, 70 Polyhedron, 70 Polynomial complexity: definition, 394 genuine (strong), 396, 450, 617 issues, 39 Polynomially bounded, 394 Polynomial-time : 743 algorithm, 81,394, 396, 546 primal simplex, 512 rounding scheme, 409, 410, 442, 451 scaling algorithm, 617 Polytope, 70, 75 Positive definite matrix, 438 Post-optimality analysis, 296 Postorder traversal, 483 Potential function, 418, 420 Predecessor index, 482 Predictor-corrector algorithm, 435 Preemptive priority: approach, 195 equivalent weights, 195 Preflow-push strategy, 617, 619, 678 Preorder distance, 483 Preorder traversal, 483 Price (fair), 272 Price-directive decomposition, 339, 373,392 Price-quantity equilibrium, 39 Pricing, 121, 206 Primal: breakthrough, 594 feasibility, 239 problem, 2, 259 simplex method, 108, 121, 220 Primal^dual method: case of unbounded dual, 288 development, 286 dual of restricted primal problem, 287 Hungarian algorithm, 535, 541, 564 modifying dual solution, 287 out-of-kilter method, 567 path-following methods, 450 restricted primal problem, 286 summary, 288 tableau format, 290 Primal-dual relationships, 264 Product form of inverse, 140, 207 Production-inventory problem, 37 Production scheduling problem, 9, 31, 33 Production-transportation-inventory problem, 505 Programming problem: generalized linear, 385 large scale, 339 linear, Projective: algorithm, 450 transformation, 404 744 Project management, 669 Project selection problem, 34 Proportionality assumption, Pseudoflow, 593, 619, 679 Pseudo-polynomial, 396, 617 Pseudorooted spanning forest, 496 Purification scheme, 408, 410, 451 QR factorization, 443 Rank: of matrix, 61 of network flow matrix, 459 of node, 457 of transportation matrix, 517 Ranking extreme points, 196, 199 Ray, 66, 118 Recession: cone, 74 direction, 66, 74 Rectilinear distance, 30 Reduced assignment matrix, 538 Reduced cost coefficients, 104 Reduced matrix, 538 Redundant: algebraically, 162 constraints, 61, 71, 73 geometrically, 71, 190 Regression equation, 207, 256 Regularizing, 414 Relationships: primal vs dual objective values, 264 primal vs dual problems, 60, 265, 266 RELAX, 511 RELAX-IV, 605 Relaxation: algorithms, 593, 605 Lagrangian, 371, 373 strategy, 372, 418 Relaxed master program, 372 RELAXT, 511 Replacing vector in the basis, 50 Representation of nonbasic vector, 132, 465, 520 Representation of polyhedral sets, 75 Representation theorem: bounded sets, 75 unbounded sets, 76 Requirement space, 22, 23, 25 Rerooting, 487 Residual capacities, 595, 618 Resolution theorem, 77 Index Resource-directive decomposition, 373, 392 Restricted primal problem, 286 Restriction strategy, 405, 418 Restrictions, see Constraint Return arc, 608 Reverse arcs of a cut, 609 Reverse cut, 609 Reverse shortest path problems, 679 Reverse-star, 455 Reverse thread index, 483 Revised simplex method: comparison with simplex method, 205 summary, 201 tableau, 202 Right-hand derivative, 604 Right-hand shadow price, 275 Right-hand-side column, 126 Right-hand-side vector, 2, 126 RNET, 511,605,678 Robust shortest path problems, 679 Rocket launching problem, 35 Root arc (root), 461 Root node, 456, 461 Rooted spanning forest, 463, 504 Rooted spanning subgraph, 462 Rooted spanning tree, 461, 518, 650 Rounding scheme, 409, 410, 442, 451 Row generation technique, 372 Row operations, 54 Row rank, 61 Row vector, 45 Row zero, 126 Saturated arc, 594, 665 Scan eligible list, 628 Scalar multiplication: of matrix, 52 of vector, 46 Scaling, 219 Scaling method, 617, 618 Schwartz inequality, 47 Self-dual algorithm, 333 Sensitivity analysis: addition: new activity, 301 new constraint, 302 bounded variables case, 332 change in: constraint matrix, 298 cost vector, 296 right-hand-side vector, 297 tolerance approach, 308 Index bounded, 20, 75 convex, 64 operations, 29 Shadow prices, 271, 273 under degeneracy, 273 right-hand/left-hand, 275 via parametric analysis, 318 Sharp labels, 630 Shortest path: algorithm: arbitrary costs, 625, 630, 633, 635 nonnegative costs, 620 approach-dependent, 639, 679 arborescence, 622 bicriteria, 679 complexity analysis, 622, 630, 633 dynamic problems, 679 dynamic stochastic problems, 639, 679 extensions, 639 flow augmentations, 679 interpretation of primal simplex, 622, 632 label-constrained, 639, 679 multi-criteria, 679 partitioned method, 635 polynomial-time algorithms, 635 problem, 547, 600,617,679 reverse path problem, 679 robust problem, 679 stochastic, 639, 679 time-dependent, 639, 679 time-restricted reverse problem, 680 threshold partitioned method, 637, 679 tree, 622, 637, 679 Signature algorithm, 565 Simple path, chain, circuit, cycle, 456 Simplex method: bounded, 220, 230 column method, 250 complexity, 397 dual, 279 finite convergence, 122, 181, 285 implementation remarks, 180, 219 initial solution, 129, 151, 293, 475, 489, 522 interpretation through KKT conditions, 242 network, 466 optimality criterion, 104, 225, 280 745 pivoting, 127 polynomial-time, 512 primal, 108, 121 primal-dual, 288 revised, 202 tableau format, 125, 129 transportation, 522 Simplex multipliers, 121, 267 Simplex path, 107 Simplex set: definition, 107 volume, 437 Simplex tableau: bounded, 225 dual, 279 interpretation of entries, 131 network, 481 primal, 125 transportation, 535 Simultaneous equations, see System of linear equations Single artificial constraint technique, 293 Single artificial variable technique, 173, 198 Single commodity flow problems, 639 Single node ascent step, 605 Single node breakthrough, 594 Singular matrix, 56 Sink, 454 Size of an instance, 394 Skew-symmetric matrix, 53, 335 Slack variable, Sliding objective method, 424 Solution: basic feasible, 62, 95 efficient, feasible, optimal, 18 Pareto-optimal, space, 18 Solvers, commercial, 257 Source, 454 Space: Euclidean, 48 requirement, 22, 23, 25 solution, 18 Spanning: forest, 457 set, 49 subgraph, 456 tree, 456, 518 Sparse, 206, 219 Special structures, 339 746 Spikes, 219 Stabilized column generation, 344, 374,391,392 Stage, 181,491 Staircase structure, 384 Stalling: definition, 175, 180, 181 in networks, 488, 493 prevention rules, 491 stages, 181,491 Standard form: of dual, 260 of linear program, 5, State variable, Steepest edge selection, 219, 256 Stepping stone method, 564 Street distance, 30 Strict complementary slackness property, 336 Strict convex combination, 64 Stroke encoding, 396, 617 Strong duality, 266 Strongly feasible basic partition, 182, 229, 488, 489 Strongly feasible trees, 489, 491, 564 Strongly (genuinely) polynomial, 396, 450,617 Strong theorem of complementary slackness, 336 Structural: constraints, variables, Subgraph: definition, 456 induced by node set, 456 maximal connected, 456 proper, 456 spanning, 456 Submatrix, 53 Suboptimization, 220, 256 Subproblem, 339, 342, 356, 363, 372 Subtree rooted at a node, 482 Successive linear approximation, Successive shortest path algorithm, 546, 562 Successor nodes, 482 Sum vector, 46 Supervisor's principle, 268 Supporting hyperplane, 88 Surplus variable, Symmetric matrix, 53 Synthesis problem, 607, 654, 680 System of linear equations: basic solution, 62 Index dependent, 61 Gaussian reduction, 56, 63 Gauss-Jordan reduction, 56, 62, 212 general solution, 62 number of solutions, 62 redundant, 61 solving, 55 Tableau: assignment, 538 bounded simplex, 225 dual simplex, 279 primal-dual, 290 revised simplex, 202 simplex, 125,481,535 transportation, 514 two-phase, 164 Tangential approximation method, 374 Tangential support, 374 Tanker scheduling problem, 13 Technological coefficients and constraints, Termination criterion: infeasibility, 21, 22, 24, 155, 169, 170,280,288 optimality, 18, 19, 114, 169, 242, 280, 288 unboundedness, 21, 27, 92, 117, 169,231,280,288 Theorems of the alternative, 235 Thread index, 482 Threshold partitioned shortest path algorithm, 637, 679 Tie breaking rule, 178 Tight constraint, binding, active, 71 Time-space network representation, 638 Tolerance sensitivity approach, 308 To-node, 455 Totally unimodular matrix, 463, 464, 517 Tourism problem, 332 Traffic assignment problem, 37, 675 Transposition of matrix, 53 Transportation problem: algorithm, 522 balanced, 514 characterization of basis for, 518 corners of cycle in tableau, 517, 520 definition, 11,513 degeneracy in, 528, 534 properties of constraint matrix, 516 Index rank of matrix, 517 representation of nonbasic vector, 520 simplex tableau associated with, 535 starting solution, 522, 558 tableau, 514 Transshipment: node, 454 problem, 551, 562 Traveling salesman problem, 501 Tree: correspondence to basis in network flows, 461,463, 518 definition, 456 dominant requirement, 655, 656 end node, 457, 518 equivalent characterizations, 458 maximum spanning tree, 656 one-, 458 properties, 457, 458 rooted, 461, 518, 650 spanning, 456, 518, 650 strongly feasible, 489, 491, 564 uniform requirement, 661 Triangularity of basis in network flows, 463,518 Triangularization, 212 Triangular matrix, 53 Two-person zero-sum game, 324 Two-phase method: analysis, 157 comparison with big-M method, 168 description, 154 Two-stage stochastic program with recourse, 39 Unary encoding, 396, 617 Unbounded: optimal value, 21, 27, 92, 117, 169, 231,265,267,280,288 polyhedral set, 21, 76 subproblem region in decomposition algorithms, 354 Uncapacitated problem, 454 Undirected graph, 455, 654 Uniform requirement tree, 661 Unimodularity, 517 Unique: optimal solution, 20, 114 solution, 62 Unit vector, 46 Unrestricted variables, 4, 147 747 Updated: basis inverse, 208, 307 column vector, 209, 342 LU factors, 217 network list structures, 485 tableau, 128, 227, 228, 307 Upper bounds on variables, 220, 478 Upper Hessenberg matrix, 218 Upper tree, 488 Upper triangular, 53 Valid cut, 304 Valid inequality, 304 Variable: artificial, 153, 198,522 basic, 95 blocking, 106, 112 bounded, 220, 478 definition, dual, 239, 260, 269 entering basis, 106, 121, 178,225, 493 flow, 454 integer, 304 leaving basis, 112, 121, 178,479, 527 legitimate, 153 nonbasic, 95 slack, surplus, unrestricted, 4, 147 Variable splitting technique, 511 Variable upper bounding, 257 Vector: basic, 49 definition, 45 dependent, 49 direction, 66, 71 eta, 208 Euclidean space, 48 independent, 48 lexicographically nonnegative, 182 lexicographically positive, 182 norm, 47 normal, 48, 67 operations, 46 sum, 46 unit, 46 zero, 46 Vertex, 72 Vertex of ray, 66, 118 Vogel's approximation method, 524, 558 Volume of simplex, 437 Index 748 Warehouse location problem, 388 Weak duality, 264 Weak theorem of complementary slackness, 268 Weighted average, 64 What-if analysis, 295 Working basis, 221 Zero: matrix, 52 vector, 46 Zero-arcs, 545 Zero-sum game, 324 ... solution and its sensitivity to relevant system parameters, and studies its predictions to various what-if types of scenarios This analysis provides insights into the system One can also use this analysis... Functions 2.5 Polyhedral Sets and Polyhedral Cones 2.6 Extreme Points, Faces, Directions, and Extreme Directions of Polyhedral Sets: Geometric Insights 2.7 Representation of Polyhedral Sets Exercises... characteristics of network structured linear programming problems and discuss the specialization of the simplex algorithm to solve these problems A detailed discussion of list structures, useful